We have just one component and suppressed the spin and polarization information for convenience. The dispersion relation \(E_{k} = \sqrt{c^2 \mathbf {k}^2 + m^2 c^4}\)  bears the fact that the theory is Lorentz covariant. Written in real space, the integrand in the Hamiltonian will involve spatial derivatives in fields, resulting in to non-commuting Hamiltonian densities. But, a direct consequence of Carroll invariance is commuting Hamiltonian densities , as per [1]}. Therefore it follows that theories with no spatial derivative in real space Hamiltonian and hence having a non-dispersive Hamiltonian in terms of Fourier modes should have Carrollian symmetry. We have also seen that there are Carrollian theories having spatial derivatives, like in the case of our upper fermions. But still the Hamiltonian densities commute and there is emergent Carrollian symmetry.
